In fact, almost everything on the number line is non-computable.
This statement is rather interesting. When you are looking at the different values that lie on the number line between two different values, you can never truly pinpoint a value because you can always attempt to make the decimal more precise. I also really like the line "In mathematics, you don't understand things. You get used to them."
And yet there are lots of numbers that aren’t fractions, perhaps the most famous being the square root of 2.
I feel like this fact is a never ending debate in mathematics. It is very interesting to see what mathematicians have to say on either side of the argument.
Most calculus students would be hard pressed to say exactly what these numbers are. They recognizethatRincludes all of IN,Z, andQand also many new numbers, such as√2,e, andπ.But asked whata real number is, many would return a blank stare. Even just asked what√2,e, orπareoften producespuzzlement. Well,√2 is a number whose square is 2. But is there a number whose squareis 2? A calculatormight oblige with 1.4142136, but(1.4142136)2̸=2
I notice that this is something we went over in my MATH 262 class in Spring 2016. My class spent a solid 20 minute on the debate that the square root of 2 as a decimal multiplied by itself was not equal to 2 because the decimal is often simplified in a calculator. I also notice that the author mentions how students know that the real numbers include all the rational and irrational numbers as well as the natural numbers, but they are never entirely sure on what defines a real number. A real number R is defined as a value that represents a quantity along a number line. Any number that can be defined as a point on a number line is considered a real number.
Ultimately the students must learn that
The equation that follows this statement (m/n + a/b = mb + na/nb) is a little confusing to follow. I am unsure of how you get from one side of the equation to the other. My algebra skills may also be a little rusty and I am just not seeing it right away, but I could use some clarification.
Many manipulations of sets require two or more operations to be performed together.The simplest cases that should perhaps be memorized areA\(B1∪B2)=(A\B1)∩(A\B2)and a symmetrical versionA\(B1∩B2)=(A\B1)∪(A\B2).If you sketch some pictures these two rules become evident
I am still a little confused on the idea of De Morgan's Laws. I get the basics, but it is still a little confusing. I wonder if this section would be a little easier to understand if the pictures the author says can be sketched were shown in the text. Sometimes my mind understands material better if there is a diagram shown along with the statement. These statements however appear to be true based on the commutative property. When I attempted to sketch out the cases, they appeared to be symmetrical to each other.
Setsare just collections of objects. In the beginning we are mostly interested in sets of real numbers.
I notice that this term was used a lot in previous classes, like MATH 180 and we started to look at in more closely in the first day of my MATH 301 class. Sets are basically the way we group and gather numbers/different types of numbers together. The most commonly seen sets are the sets of real numbers, integers, rational numbers, irrational numbers and complex numbers. Sets can also be divided into subsets, which are smaller groups of elements that belong to the same set.
From the iron grate example, we know how to map A, B, and C to T(A), T(B), and T( C), respectively, by a sequence of at most three reflections.
I feel that we would benefit more from seeing this "iron grate" example and how it shows up the proper mapping for A, B, and C, as well as diagrams of their three reflections. It is harder for me to follow along with just the words than it is with diagrams to look at.
As an example of the difference in the meaning of this language, consider the mapping of an equilateral triangle ABC onto itself by a rotation of 120° around the centroid. The triangle ABC is fixed by the transformation, but it is not fixed pointwise.
I feel that this would be easier to understand if we were given a diagram to follow along with as well as a more detailed explanation.
We want to show that T(X) = X for every point X in the plane. Suppose that this is
I'm confused by what the author is saying here. We want to show that T(X) = X for every X in the plane, but then says that we are supposed to say this is not the case. How are we expected to prove this idea if he's mixing the ideas like this?
If it is also true that the commutative law holds (that is, if a · b = b · a for all a and b in Q), then g is called an Abelian group.
This ties into what I learned in Abstract Algebra about Abelian groups. An abelian group is defined as a group in which the result of applying the group operation to two group elements does not depend on the order they are written in because they are commutative. Does this mean that all isometries are also Abelian groups?
For example, we could create a new isometry by first doing a rotation, then a reflection about some line, then a reflection about another line, then a translation.
Is it possible to see an example of this? It is hard to understand how this is possible without an example to follow/prove the point.
We must check two things: (a) that every point R on l has its image R' on m and (b) that every pointS' on m has its preimage Son l.
How are we supposed to check these facts? It appears that the author simply rushes to the end rather than take the time to show us how to check each of these to help clarify the proof.
EXAMPLES 69 By Pythagoras' Theorem, AB
(I apologize for my highlighting; it is difficult on and iPad) I do not understand how the Pythagoras's Theorem is used here. The author gives no insight to how they formed these equations were created, nor how they actually make the theorem true.
equivalently, tha
This section could use a little more explanation to it. Based on how the equation is set up, you can tell that the author used crossed multiplication, but how did he get the ratios he did in order to perform this operation?
Then it follows that AB AC AP AQ' and subtracting 1 from both sides of the
This piece looks very familiar to 3.3.4 (sss Similarity). It uses proportions rather than congruent sides to prove similarity between the triangles. So we can see that since PQ and BC are parallel to each other, we can use Theorem 3.2.4 to prove similarity of the triangles.
There are other conditions that allow us to conclude that two triangles are similar, but there
Didn't we go over another theorem or lemma that proved the similarity of triangles? What makes this one better to use?